# Circle A has a center at #(-4 ,6 )# and a radius of #4 #. Circle B has a center at #(1 ,1 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?

The circles do not overlap.

Smallest distance between them is 4

If you find the distance between centres then directly compare it to the sum of the radii you can determine if they do overlap or not.

Using Pythagoras

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no overlap , distance ≈ 1.071

now : radius of A + radius of B = 4+2 =6

since sum of radii < distance between centres → no overlap

distance between circles = 7.071 - 6 = 1.071

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To determine if the circles overlap, we need to find the distance between their centers and compare it to the sum of their radii.

The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the distance formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).

For Circle A with center ((-4, 6)) and Circle B with center ((1, 1)):

Distance between centers: [d = \sqrt{(1 - (-4))^2 + (1 - 6)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}]

Sum of radii: (r_A + r_B = 4 + 2 = 6)

Since the distance between the centers ((5\sqrt{2})) is greater than the sum of the radii (6), the circles do not overlap.

The smallest distance between the circles is the difference between the distance between their centers and the sum of their radii: [5\sqrt{2} - 6]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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